Let \(v \in \mathbb {P}_p(\tau )\) , let \(F\)  be a face of \(\tau \)  and set
\({n}_F:={n}_{\tau |F}\) . Then \(w := \nabla v \cdot {n}_F \in \mathbb {P}_{p-1}(\tau )\) .
Applying the discrete trace inequality from [1]} yields
\(\Vert  w \Vert _{L^2(F)}^2 \le p(p+d-1) \frac{|F|}{d|\tau |} \Vert  w \Vert _{L^2(\tau )}^2.\) 
